Numerical experiments on the convergence properties of state-based peridynamic laws and influence functions in two-dimensional problems

Abstract Peridynamics is widely used as the theoretical basis for numerical studies of fracture evolution, propagation, and behavior. While the theory has been shown to converge to continuum mechanics in the theoretical limit, its behavior as a discrete numerical approximation with respect to classic problems has not been shown. In this study, we use standard analytical solutions to thoroughly test the numerical accuracy and rate of convergence of the spatial discretization obtained by peridynamics. We analyze the accuracy and rate of convergence of three different peridynamic constitutive responses: of these, two involve a state-based dilation, and the third is based on the estimation of the deformation gradient. Additionally, we study the choice of the peridynamic influence function in each of the constitutive responses. The peridynamic materials are solved in the linear elastic regime by solving a linear system of equations obtained by symbolically differentiating the force states. We test the methods against standard constant-strain solutions for uniaxial compression, isotropic compression, and simple shear. We also apply the methods to a finite material with a pressurized thin crack, using the Westergaard’s solution method to obtain an analytical displacement field for comparison. The two dilation-based peridynamic constitutive responses are found to only converge to one of the constant strain solutions, while the deformation gradient-based law converges in all cases with an appropriate choice of the influence functions. We show that a cubic influence function is the best choice of those considered in all methods. Only the deformation gradient-based model converges for all three linear deformation problems, but is less accurate than the dilation-based models for the thin crack problem because of instabilities. We demonstrate an ad hoc smoothing technique based on the influence function that is able to alleviate these instabilities and improve the accuracy of the deformation gradient-based model.

[1]  Erkan Oterkus,et al.  Peridynamic analysis of fiber-reinforced composite materials , 2012 .

[2]  S. Silling,et al.  Peridynamic States and Constitutive Modeling , 2007 .

[3]  Gaël Varoquaux,et al.  The NumPy Array: A Structure for Efficient Numerical Computation , 2011, Computing in Science & Engineering.

[4]  Erkan Oterkus,et al.  Peridynamic Theory and Its Applications , 2013 .

[5]  Michael L. Parks,et al.  ON THE ROLE OF THE INFLUENCE FUNCTION IN THE PERIDYNAMIC THEORY , 2011 .

[6]  John D. Hunter,et al.  Matplotlib: A 2D Graphics Environment , 2007, Computing in Science & Engineering.

[7]  Richard A. Regueiro,et al.  Peridynamics simulations of geomaterial fragmentation by impulse loads , 2015 .

[8]  S. Silling,et al.  Viscoplasticity using peridynamics , 2010 .

[9]  Daniel Z. Turner,et al.  A Non-Local Model for Fluid-Structure Interaction with Applications in Hydraulic Fracturing , 2012, ArXiv.

[10]  Selda Oterkus,et al.  Peridynamic thermal diffusion , 2014, J. Comput. Phys..

[11]  Steven J. Plimpton,et al.  Implementing peridynamics within a molecular dynamics code , 2007, Comput. Phys. Commun..

[12]  F. Bobaru,et al.  The Meaning, Selection, and Use of the Peridynamic Horizon and its Relation to Crack Branching in Brittle Materials , 2012, International Journal of Fracture.

[13]  Ted Belytschko,et al.  A meshfree unification: reproducing kernel peridynamics , 2014, Computational Mechanics.

[14]  Stewart Andrew Silling,et al.  Crack nucleation in a peridynamic solid , 2010 .

[15]  Mukul M. Sharma,et al.  A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamics approach , 2015 .

[16]  Jung-Wuk Hong,et al.  Discretized peridynamics for linear elastic solids , 2012 .

[17]  Erkan Oterkus,et al.  Peridynamic theory for modeling three-dimensional damage growth in metallic and composite structures , 2010 .

[18]  I. N. Sneddon.  The distribution of stress in the neighbourhood of a crack in an elastic solid , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[19]  Sandia Report,et al.  Peridynamics with LAMMPS: A User Guide v0.3 Beta , 2010 .

[20]  Erkan Oterkus,et al.  Ordinary state-based peridynamic material constants , 2012 .

[21]  S. Silling,et al.  Convergence, adaptive refinement, and scaling in 1D peridynamics , 2009 .

[22]  S. Silling.  Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces , 2000 .

[23]  Mukul M. Sharma,et al.  A peridynamic formulation of pressure driven convective fluid transport in porous media , 2014, J. Comput. Phys..

[24]  Akira Nishida,et al.  Evaluation of Parallel Aggregate Creation Orders: Smoothed Aggregation Algebraic Multigrid Method , 2005 .

[25]  E. Emmrich,et al.  On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity , 2007 .

[26]  Nicolas Sau,et al.  Peridynamic modeling of concrete structures , 2007 .

[27]  Eric Jones,et al.  SciPy: Open Source Scientific Tools for Python , 2001 .

[28]  Richard B. Lehoucq,et al.  Peridynamics as an Upscaling of Molecular Dynamics , 2009, Multiscale Model. Simul..

[29]  Justin Schwartz,et al.  A two‐dimensional ordinary, state‐based peridynamic model for linearly elastic solids , 2014 .

[30]  Mazdak Ghajari,et al.  A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media , 2014 .

[31]  An Assessment of Peridynamics for Pre and Post Failure Deformation , 2011 .

[32]  Youn Doh Ha,et al.  ADAPTIVE REFINEMENT AND MULTISCALEMODELING IN 2D PERIDYNAMICS , 2011 .

[33]  R. Lehoucq,et al.  Convergence of Peridynamics to Classical Elasticity Theory , 2008 .

[34]  S. Silling,et al.  A meshfree method based on the peridynamic model of solid mechanics , 2005 .